Autocatalytic Step Research Articles

Autocatalytic cooperativity and self-regulation of ATPase pumps in membrane active transport.

We investigate the effect of autocatalysis on the conformational changes of membrane pumps during active transport driven by ATP. The translocation process is described by means of an alternating access model. The usual kinetic scheme is extended by introducing autocatalytic steps and allowing for dynamic formation of enzyme complexes. The usual features of cooperative models are recovered, i.e., sigmoid shapes of flux versus concentration curves. We show also that two autocatalytic steps lead to a mechanism of inhibition by the substrate as experimentally observed for some ATPase pumps. In addition, when the formation of enzyme complexes is allowed, the model exhibits a multiple stationary states regime, which can be related to a self-regulation mechanism of the active transport in biological systems.

Chaotic behavior of reaction systems: consecutive quadratic/cubic autocatalysis via intermediates

The chaotic behavior produced by a reaction system composed of consecutive quadratic and cubic autocatalytic elementary steps has been described. It is shown that the existence of chaotic behavior does not depend on the presence of a reaction step involving cubic autocatalysis because chaotic behavior is preserved when the cubic step is replaced by successive bimolecular steps involving an intermediate. The chaotic behavior, as characterized by the Lyapunov exponents, displayed by the model based on the bimolecular approximation approaches, in the limit, that of the model based on cubic autocatalysis. Chaos also occurs for the bimolecular approximation model for situations in which the limiting situation of cubic autocatalysis is not approached.

Existence and uniqueness of periodic orbits in the cubic autocatalator

Periodic orbits are sought in a mathematical model of a simple prototype chemical reaction involving essentially only two reacting species. Physically, these periodic orbits correspond to time-periodic oscillations in the concentrations of the two chemicals. Using the results for the existence and uniqueness of periodic orbits for Lienard systems, necessary and sufficient conditions are obtained for the existence of exactly one periodic orbit and of no periodic orbits. The results apply to a closed system where the quadratic autocatalytic reaction and decay step are present but the uncatalysed reaction is not and where there is only one physically relevant equilibrium solution.

A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay

In this note we consider a reaction-diffusion problem which describes a simple model chemical reaction scheme for quadratic autocatalysis with linear decay. We show that withk>1 (wherek is a parameter measuring the relative strength of the decay step to the autocatalytic step) the dimensionless unreacting state α≡1, β≡0 is globally asymptotically stable, with α∼1+0(t −1/2) and β∼ 0(t −1/2 e −(k−1)t) ast→-∞. Here α and β are the concentrations of the reactant and the autocatalyst respectively, andt is time. The casek<1 has been considered in detail by Merkin et al. [1].

The effect of flow and mass transport in thrombogenesis

The paper presents a mathematical analysis of the contributions of flow and mass transport to a single reactive event at a blood vessel wall. The intent is to prepare the ground for a comprehensive study of the intertwining of these contributions with the reaction network of the coagulation cascade. We show that in all vessels with local mural activity, or in "large" vessels (d greater than 0.1 mm) with global reactivity, events at the tubular wall can be rigorously described by algebraic equations under steady conditions, or by ordinary differential forms (ODEs) during transient conditions. This opens up important ways for analyzing the combined roles of flow, transport, and coagulation reactions in thrombosis, a task hitherto considered to be completely intractable. We report extensively on the dependence of transport coefficient kL and mural coagulant concentration Cw on flow, vessel geometry, and reaction kinetics. It is shown that for protein transport, kL varies only weakly with shear rate gamma in large vessels, and not at all in the smaller tubes (d less than 10(-2) mm). For a typical protein, kL approximately 10(-3) cm s-1 within a factor of 3 in most geometries, irrespective of the mural reaction kinetics. Significant reductions in kL (1/10-1/1,000) leading to high-coagulant accumulation are seen mainly in stagnant zones vicinal to abrupt expansions and in small elliptical tubules. This is in accord with known physical observations. More unexpected are the dramatic increases in accumulation which can come about through the intervention of an autocatalytic reaction step, with Cw rising sharply toward infinity as the ratio of reaction to transport coefficient approaches unity. Such self-catalyzed reactions have the ability to act as powerful amplifiers of an otherwise modest influence of flow and transport on coagulant concentration. The paper considers as well the effect on mass transport of transient conditions occasioned by coagulation initiation or pulsatile flow. During initiation, instantaneous flux varies with diffusivity and bulk concentration, favouring the early adsorption/consumption of proteins with the highest abundance and mobility. This is akin to the 'Vroman effect' seen in narrow, stagnant spaces. The effect of flow pulsatility on kL has the potential, after prolonged cycling, of bringing about segregation or accumulation of proteins, with consequences for the coagulation process.

Linear thermodynamic analysis of the reversible Selkov model: An interpretation of the Chatelier-like principle for local concentration fluctuations near thermodynamic equilibrium

Thermodynamic analysis of the reversible Selkov model (a simple kinetic model describing glycolytic oscillations) has been done by an entropy production technique of Prigogine and it is shown that only the autocatalytic step can destabilize the steady state in this model. It is derived that at thermodynamic equilibrium, the product δS δP is always a positive quantity which appears to be a Chatelier-like principle for local concentration fluctuation applicable to the autocatalytic step S ⇄ P of this model.

Rate of the Bromous Acid-Bromide Reaction Measured in a Br2-HOBr Buffer System in Sulfuric Acid Solution

In the Belousov-Zhabotinsky system bromine species of oxidation states - 1 to 5 are important for the start and for the inhibition of the autocatalytic reaction steps. Especially, the reaction of Br- with HBrO2 competes with the oxidation of Ce3+ by BrO2 formed from HBrO2 and HBrO3. In our investigation the reaction of Br- with HBrO2 (solvent 1 M sulfuric acid) is followed spectroscopically in a mixture of bromine and hypobromous acid, which system works as a buffer for bromide. Including experiments on the hydrolysis of Br2 and on the solubility product of AgBr in 1 M sulfuric acid solution direct information about the rate constant kl of the Br-/HBrO2 reaction is obtained (k1 = 2.5 × 106 M-2 s-1 at 20 °C and 4× 106 M-2 s-1 at 25°C, which values are valid in the range of 0.01 to 1 M sulfuric acid).

On the structural stability of the cubic autocatalytic scheme in a closed vessel

This paper is concerned with the cubic autocatalator, A+2B→2B, B→C, in a closed system, in particular when the extra uncatalysed reaction A→B, the quadratic autocatalytic reaction A+B→2B and the additional decay step A→C are included. Previous work has shown that the oscillatory behaviour in the original scheme is structurally unstable with respect to the inclusion of the uncatalysed reaction. Here the equations exhibit a more complex pattern of behaviour than was present in previous studies and this has been fully investigated using constructional methods developed by Gray and Roberts. It is shown that the basic scheme is also structurally unstable with respect to the inclusion of the quadratic autocatalytic step. However, the inclusion of the extra decay step A→C has the effect of removing this structural instability by bringing into the finite phase plane behaviour that was previously seen at infinity.

Analysis of chemical kinetic systems over the entire parameter space - II. Isothermal oscillators

Two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit. The first scheme, which can be regarded as one of the simplest isothermal models of interest, is based on the ‘quadratic’ autocatalytic step A + B → 2B rate = k q ab , where a and b are the concentrations of A and B respectively, combined with a nonlinear termination step of the form B → C rate = k t b /(1+ rb ), where r is some parameter. The second scheme is a simplified (by treating one parameter as a constant) version of a model that has been proposed for the oxidation of carbon monoxide (CO). This scheme has a termination step of the same form as above, and like the ‘quadratic’ model, the highest nonlinearity apart from this term is quadratic. For the quadratic model we find 10 qualitatively distinct phase portraits and, with our choice of bifurcation parameter, 22 distinct bifurcation diagrams. The most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles. For the CO model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams. In both models much of the behaviour occurs over only extremely small regions in parameter space.

Review Lecture - Instabilities and oscillations in chemical reactions in closed and open systems

This lecture is chiefly concerned with isothermal oscillations in homogeneous systems, and with finding and validating self-consistent models. Emphasis is laid on satisfying basic physical principles and on preserving chemical reasonableness in as simple a manner as possible. The goal is to devise a scheme that can be applied to a wide variety of circumstances to predict new behaviour as well as to interpret what is known, and to form a foundation for later developments and elaborations. This core of the scheme, called the autocatalator, combines an autocatalytic step A + 2B → 3B, rate =k1ab2, with catalyst decay B → C, rate =k2b. These core reactions may display reversibility and they may be accompanied by uncatalysed conversion A → B, rate =kua. Particular stress is given to the problem of closed systems. In closed systems, oscillations cannot be indefinitely sustained. In former days it was customary to ignore this inconvenience (and to repeal the law of conservation of matter); today a less unsatisfactory account is possible.

Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations

The simplest representations of non-linear chemical feedback in an isothermal system are the prototype autocatalytic steps ▪ Such rate-laws arise in both solution-phase and gas-phase kinetics and in biological situations. In the present paper these reactions are coupled with the diffusion of the reactants through a permeable boundary from an external reservoir where the concentrations are held constant. When the autocatalyst reacts further by a first-order decay to a stable product C ▪ a wide range of complex behaviour is possible even for this simple system. The dependence of the stationary-state conversion of A on the experimental conditions may display isola or mushroom patterns (with two branches forming a closed loop or two separate ranges of hysteresis and multiplicity respectively). The stationary-state autocatalyst concentration profile may display multiple internal extrema. Sustained temporal oscillations are also possible, corresponding to stable limit cycles which emerge from Hopf bifurcation points. These are located by local stability analysis and confirmed by extensive computation.

50] Oscillatory phenomena in immobilized enzyme systems

Oscillatory phenomena have generally been observed experimentally in either homogeneous or pseudo-homogeneous media in connection with enzymes in soluble phase. However, in such systems, several enzymatic species and crossed feedback regulations are required to explain time oscillations. The same phenomena can be achieved under simpler biochemical conditions only by mass transfers. This chapter describes some experimental model systems exhibiting time oscillations that have been studied. Their common features are (1) one single operating enzyme that exhibits an autocatalytic step in its reaction term, for example, substrate inhibition and pH dependence and (2) the coupling of the enzyme reaction with the diffusion of metabolites (substrates and/or products). The reproducible manufacture of artificial membranes containing active enzymes is important in allowing a precise mathematical description, within a well-defined context, of the (oscillatory) system to be studied.

Oscillatory chemical reactions in closed vessels

A skeleton kinetic scheme representing the simplest model for oscillatory chemical reactions in a closed vessel can be built around an autocatalytic feedback step precursor decay P → A k 0 p , uncatalysed step A → B k 3 a , autocatalysis A + 2B → 3B k 1 ab 2 , catalyst decay B → C k 2 b . The first intermediate A is formed via the slow decay of a reactant or precursor species P, initially in large excess. A is converted to B via two routes: a slow pseudo-first order process and a step in which B acts as its own catalyst. The autocatalyst B is then capable of a simple first order decay to a stable product C. The concentrations of the various species at first change steadily, with that of P decreasing while A, B and C increase. This period is followed by the onset and growth of oscillations in the concentrations of the intermediates A and B. The behaviour at long times, depends upon the uncatalysed conversion of A to B. Provided k 3 is not taken as zero, the oscillations finally diminish in amplitude and die out leaving a steady decay of P, A and B until everything has been converted to C. The simplicity of the model allows the first self-consistent test of the ‘pool chemical approximation’, an approach commonly used in the analysis of mechanisms in closed systems in which the precursor concen­tration is assumed to be constant and set equal to its initial value. The results presented here reveal the range of applicability of the approxi­mation and show clearly how and why it can break down to give un­physical predictions.

ChemInform Abstract: Systematic Design of Chemical Oscillators. Part 29. Reaction Between Permanganate Ion and Hydrogen Peroxide: Kinetics and Mechanism of the Initial Phase of the Reaction.

AbstractThe MnO; /H2O2 reaction, studied by stopped‐flow experiments, occurs in three stages: a fast initial phase, an induction period, and an autocatalytic step.

A reduction scheme for explosive chemical kinetics

The role of combined fluctuations in the input and output flows is analyzed on a model for explosive kinetics presenting reactive intermediates. In the deterministic limit the system is an extension of other kinetic schemes extensively studied which have third-order autocatalytic elementary steps. However, it represents a more realistic situation since it does not involve such low-probability third-order collision processes. It has an additional fast-relaxing degree of freedom: the concentration of an intermediate species. The multiplicative noise in the evolution of the fast variable corresponds to the fluctuations in the input flow. Only if these fluctuations are neglected in the adiabatic elimination limit, our model yields the results as obtained with previous models. However, applying the center manifold techniques, we show that these fluctuations are crucial since they can reinforce or inhibit the explosive tunneling effect caused by the noise in the output flow rate.

Periodic nucleation processes in emulsion polymerization systems

Under special circumstances, Liesegang rings, which are normally associated with the periodic precipitation of inorganic salts, can be observed in emulsion polymerization systems. A quantitative theory is presented for the formation of these polymer rings, based on the numerical solution of coupled reaction-diffusion equations describing their temporal and spatial evolution. The autocatalytic step required for periodic phenomena is identified as arising from the coagulative nucleation mechanism for latex particle formation. In this mechanism, precursor particles, formed by either micellar entry or homogeneous nucleation, coagulate to give latex particles. An initial spatial inhomogeneity in the latex particle concentration grows and induces a depletion region adjacent to it. Another maximum in the latex concentration subsequently appears, separated from the initial one by a region where few latex particles have been nucleated. This type of periodic structure is shown to evolve only if the nucleation rate is an increasing function of time. Such a function is derived from extended Müller-Smoluchowski coagulation kinetics involving precursor and latex particles. Excellent agreement is obtained between the theory, using physically reasonable parameters, and the spacing between the two rings in unstirred emulsion polymerization systems.

In‐Situ IR Study during Oscillations of the Catalytic CO Oxidation

AbstractFrom earlier investigations [1, 2] it is well known that the oscillations in reaction rate of the CO oxidation on Pt or Pd run parallel with periodic changes of the CO coverage of the metal surface. This could be confirmed by following the periodic changes of the IR absorbance of the chemisorbed CO in an infrared cell with the oscillating reaction in‐situ. ‐ Spectra taken as “snap‐shots” along the oscillating period revealed that not only the overall absorbance, but also the shape of the spectra changes. The changes indicate the formation of a densely packed CO chemisorption phase (C(4 × 2) structure with θco = 0.5) besides the “normal” CO domain structure with θco = 0.33, and its removal in the cycling reaction. The removal is accomplished, presumably, by a densely packed chemisorption phase of oxygen atoms (√3 × √3/R 30° structure), that forms under compression by the dense CO phase, and is known as particularly reactive to chemisorbed CO. ‐ The formation of this aggressive chemisorption structure of oxygen, induced by the dense chemisorption phase of CO, is one autocatalytic step in this mechanism. The second non‐linear effect is the formation of CO domains at the surface during reaction by means of a precursor mechanism of the CO adsorption. These two non‐linearities are obviously the driving forces of the oscillations.

Selection and Coexistence in a Bioreactor

AbstractThe bioreactor (continuous flow stirred tank reactor) has become an ubiquitous experimental device that also allows the study of selection, bistability or coexistence of competing species in the laboratory. The steady state solutions (and relaxation times of two mechanisms) of a number of simple mechanisms are calculated that contain two competing autocatalytic steps followed by a third step. For simplicity the third step is chosen to be either a first order conversion between the rival species (mechanism II), an additional autocatalytic conversion (mechanism III), or an annihilation step between the two species (mechanism IV). Selection may be displayed by all mechanisms studied, coexistence only by mechanisms II and III, whereas IV shows bistability where either one of two steady states are reached depending upon initial concentrations. A number of relevant biological systems are described and qualitatively compared with theory.

Mechanismus der autokatalytischen Primärreaktion bei der Formose-Reaktion / Mechanism of the Autocatalytic Primary Reaction Step in the Formose Reaction

Abstract New insights into the autocatalytic primary reaction step of the formose reaction are obtained by using soluble calcium hydroxide bismethylene glycolate complexes for kinetic investigations in homogeneous phase. The reaction kinetics follows between 20% and 80% formaldehyde consumption a nonlinear rate equation. Specific co-catalytic activities are determined for the most important intermediates and end products of the formose reaction. By correlation of the kinetic results with solubility experiments, ultraviolet, 13C NMR spectroscopic measurements and rates of oxidation and acid formation, a mechanism for the autocatalytic primary reaction is given. The main catalytic effects for the formaldehyde activation are caused by calcium enediolate complexes in an autocatalytic cycle and competition with the Cannizzaro and oligomerisation reaction. Further, but less probable mechanisms are also discussed.

A novel possibility to clarify the role of bromide in bromate oscillators

Bromocomplex forming metal ions (Tl3+, Hg2+) alter considerably the parameters of chemical oscillations in bromate oscillators. Even at 10−5 M metal ion concentration the period of the oscillation is doubled or tripled; at 10−4 M the oscillation is quenched. At high (≥10−2 M) Tl3+ concentration, high frequency, strongly damped oscillations occur although [Br−]eq is below 10−10 M. By the addition of Tl3+ we could determine the amount of Br− formed in a single autocatalytic step, which we found to be about 10−4 M.